The value of genomic relationship matrices to estimate levels of inbreeding

Abstract

Background: Genomic relationship matrices are used to obtain genomic inbreeding coefficients. However, there are several methodologies to compute these matrices and there is still an unresolved debate on which one provides the best estimate of inbreeding. In this study, we investigated measures of inbreeding obtained from five genomic matrices, including the Nejati-Javaremi allelic relationship matrix (F_NEJ), the Li and Horvitz matrix based on excess of homozygosity (F_L&H), and the VanRaden (methods 1, F_VR1, and 2, F_VR2) and Yang (F_YAN) genomic relationship matrices. We derived expectations for each inbreeding coefficient, assuming a single locus model, and used these expectations to explain the patterns of the coefficients that were computed from thousands of single nucleotide polymorphism genotypes in a population of Iberian pigs.

Results: Except for F_NEJ, the evaluated measures of inbreeding do not match with the original definitions of inbreeding coefficient of Wright (correlation) or Malécot (probability). When inbreeding coefficients are interpreted as indicators of variability (heterozygosity) that was gained or lost relative to a base population, both F_NEJ and F_L&H led to sensible results but this was not the case for F_VR1, F_VR2 and F_YAN. When variability has increased relative to the base, F_VR1, F_VR2 and F_YAN can indicate that it decreased. In fact, based on F_YAN, variability is not expected to increase. When variability has decreased, F_VR1 and F_VR2 can indicate that it has increased. Finally, these three coefficients can indicate that more variability than that present in the base population can be lost, which is also unreasonable. The patterns for these coefficients observed in the pig population were very different, following the derived expectations. As a consequence, the rate of inbreeding depression estimated based on these inbreeding coefficients differed not only in magnitude but also in sign.

Conclusions: Genomic inbreeding coefficients obtained from the diagonal elements of genomic matrices can lead to inconsistent results in terms of gain and loss of genetic variability and inbreeding depression estimates, and thus to misleading interpretations. Although these matrices have proven to be very efficient in increasing the accuracy of genomic predictions, they do not always provide a useful measure of inbreeding.

Fig. 1

Expected inbreeding coefficient based on excess of homozygosity ($F_{L\&H}$) (a) and expected inbreeding coefficients computed from the diagonal elements of the genomic relationship matrices of VanRaden (methods 1 and 2; $F_{\mathit{VR}}$ = $F_{VR1}$  = $F_{VR2}$) (b) and of Yang ($F_{\mathit{YAN}})$ (c) as a function of starting and current allele frequencies at a single locus. On the right, the grid of initial and current frequencies is divided in regions where the expected value of $F$ is < − 1, < 0, between − 1 and 0, between 0 and 1, or > 1

Fig. 2

Expected $F_{L\&H}$, $F_{\mathit{VR}}(F_{VR1}=F_{VR2})$ and $F_{\mathit{YAN}}$ in the generation in which the reference allele was lost (a), driven to a frequency of 0.5 (b), or fixed (c) relative to the initial frequency ($p_{(0)}$). $F_{L\&H}$: blue line, $F_{\mathit{VR}}$: brown line, $F_{\mathit{YAN}}$: red line

Fig. 3

Scatter plots for inbreeding coefficients $F_{\mathit{NEJ}}$, $F_{L\&H}$, $F_{VR1}$, $F_{VR2}$ and $F_{\mathit{YAN}}$ in the Guadyerbas population when computed at the individual animal (a) or genomic region (b) level in cohorts 1 (left panels) and 6 (right panels), and the corresponding correlation coefficients ($r$)

Fig. 4

Evolution of the proportion of the genome that becomes homozygous (i.e. F_NEJ) from cohort 1 (grey lines) to cohort 6 (black lines) for the different chromosomes (SSC) in the Guadyerbas population when using SNPs with non-zero minor allele frequencies. The horizontal lines represent averages across the genome

Fig. 5

Patterns of different measures of genomic inbreeding ($F_{L\&H}$ blue line, $F_{VR1}$ light brown line, $F_{VR2}$ dark brown line, $F_{\mathit{YAN}}$ red line) in cohort 6 for different chromosomes (SSC) in the Guadyerbas population when using SNPs with non-zero minor allele frequencies in cohort 1

Fig. 6

Patterns of different measures of genomic inbreeding ($F_{L\&H}$ blue line, $F_{VR1}$ light brown line, $F_{VR2}$ dark brown line, $F_{\mathit{YAN}}$ red line) in cohort 6 for chromosomes 1, 4, and 17 in the Guadyerbas population when using SNPs with minor allele frequencies > 0.05 (a) or > 0;25 (b) in cohort 1

Fig. 7

Patterns of the rate of inbreeding depression (b) for number of piglets born alive in the Guadyerbas population when computed using different measures of genomic inbreeding ($F_{L\&H}$ blue line, $F_{VR2}$ brown line, $F_{\mathit{YAN}}$ red line) for specific regions of six chromosomes. All genotyped sows with phenotypic data that were born from cohort 1 to cohort 6 were included in the analyses